construct a Sylow subgroup of a group
SylowSubgroup( p, G )
a positive rational prime
a permutation group or Cayley table group
Let G be a finite group, and let p be a positive (rational) prime. A Sylow p-subgroup of G is a maximal p-subgroup of G where, by a p-subgroup, we mean a subgroup whose order is a power of p. The Sylow theorems assert that, for a prime divisor p of the order of a finite group G, there is a Sylow p-subgroup of G and that all Sylow p-subgroups of G are conjugate in G. Moreover, the number of Sylow p-subgroups of G is congruent to 1 modulo p.
The SylowSubgroup( p, G ) command constructs a Sylow p-subgroup of a group G. The group G must be an instance of a permutation group or a Cayley table group.
Note that, if p is not a divisor of the order of G, then the trivial subgroup of G is returned.
G ≔ AlternatingGroup⁡4
P2 ≔ SylowSubgroup⁡2,G
G ≔ CayleyTableGroup⁡Symm⁡5
G≔ < a Cayley table group with 120 elements >
P ≔ SylowSubgroup⁡3,G
P≔ < a Cayley table group with 3 elements >
N ≔ Normaliser⁡P,G
N≔N < a Cayley table group with 120 elements > ⁡ < a Cayley table group with 3 elements >
Q ≔ SylowSubgroup⁡2,N
Q≔ < a Cayley table group with 4 elements >
The GroupTheory[SylowSubgroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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